Lemma 2.6 ([1]}, p.539)   Let \(\Phi \)  be a function on \(\mathbb {R}^d\)  satisfying for some \(0<C,\delta <\infty ,\)  \(|\Phi (x)|\le C {(1+|x)|)^{-d-\delta }}\) . Then a measure \(\mu \)  on \(\mathbb {R}_+^{d+1}\)  is a Carleson if and only if for every \(p\)  with \(1<p<\infty \)  there is a constant \(C_{p,d,\mu }\)  such that for all \(f\in L^p(\mathbb {R}^d)\)  we have
\(\int _{\mathbb {R}_+^{d+1}} |\Phi _t*f(x)|^p d\mu (x,t)\le C_{p,d,\mu }\int _{\mathbb {R}^d}|f(x)|^pdx.\) 

